# Properties

 Label 546.2.cg.b Level $546$ Weight $2$ Character orbit 546.cg Analytic conductor $4.360$ Analytic rank $0$ Dimension $40$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$546 = 2 \cdot 3 \cdot 7 \cdot 13$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 546.cg (of order $$12$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.35983195036$$ Analytic rank: $$0$$ Dimension: $$40$$ Relative dimension: $$10$$ over $$\Q(\zeta_{12})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

## $q$-expansion

The dimension is sufficiently large that we do not compute an algebraic $$q$$-expansion, but we have computed the trace expansion.

 $$\operatorname{Tr}(f)(q) =$$ $$40q + 4q^{7} + 20q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$40q + 4q^{7} + 20q^{9} + 8q^{11} - 20q^{12} + 4q^{14} - 40q^{16} + 16q^{17} + 4q^{19} + 4q^{21} - 4q^{22} - 24q^{25} + 8q^{26} - 4q^{28} - 12q^{29} - 8q^{33} - 8q^{34} - 32q^{35} + 40q^{37} + 8q^{38} + 20q^{39} + 20q^{41} + 12q^{42} - 24q^{43} - 4q^{44} + 32q^{46} + 4q^{47} + 32q^{49} - 16q^{50} + 24q^{51} + 16q^{52} - 4q^{53} + 24q^{55} + 12q^{56} + 8q^{57} + 12q^{58} + 24q^{59} + 60q^{61} + 32q^{62} - 4q^{63} + 44q^{65} - 12q^{67} - 8q^{69} - 24q^{70} - 28q^{71} + 60q^{73} - 40q^{74} - 72q^{75} + 4q^{76} - 12q^{77} + 4q^{78} - 20q^{81} - 24q^{82} + 60q^{83} - 8q^{84} - 4q^{85} - 20q^{86} - 36q^{89} - 16q^{92} - 24q^{93} - 48q^{97} - 88q^{98} + 4q^{99} + O(q^{100})$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
145.1 −0.707107 + 0.707107i −0.866025 0.500000i 1.00000i −4.23240 + 1.13407i 0.965926 0.258819i 1.34853 + 2.27629i 0.707107 + 0.707107i 0.500000 + 0.866025i 2.19085 3.79467i
145.2 −0.707107 + 0.707107i −0.866025 0.500000i 1.00000i −1.30432 + 0.349490i 0.965926 0.258819i −1.83449 1.90648i 0.707107 + 0.707107i 0.500000 + 0.866025i 0.675164 1.16942i
145.3 −0.707107 + 0.707107i −0.866025 0.500000i 1.00000i −0.981662 + 0.263036i 0.965926 0.258819i 2.09303 1.61840i 0.707107 + 0.707107i 0.500000 + 0.866025i 0.508146 0.880134i
145.4 −0.707107 + 0.707107i −0.866025 0.500000i 1.00000i 3.01304 0.807342i 0.965926 0.258819i 2.56888 + 0.633142i 0.707107 + 0.707107i 0.500000 + 0.866025i −1.55967 + 2.70142i
145.5 −0.707107 + 0.707107i −0.866025 0.500000i 1.00000i 3.50534 0.939253i 0.965926 0.258819i −2.61012 0.432735i 0.707107 + 0.707107i 0.500000 + 0.866025i −1.81450 + 3.14280i
145.6 0.707107 0.707107i −0.866025 0.500000i 1.00000i −2.31481 + 0.620250i −0.965926 + 0.258819i −2.57277 + 0.617120i −0.707107 0.707107i 0.500000 + 0.866025i −1.19823 + 2.07540i
145.7 0.707107 0.707107i −0.866025 0.500000i 1.00000i −1.42776 + 0.382567i −0.965926 + 0.258819i 0.349274 + 2.62260i −0.707107 0.707107i 0.500000 + 0.866025i −0.739062 + 1.28009i
145.8 0.707107 0.707107i −0.866025 0.500000i 1.00000i 0.499350 0.133800i −0.965926 + 0.258819i 0.430450 + 2.61050i −0.707107 0.707107i 0.500000 + 0.866025i 0.258483 0.447705i
145.9 0.707107 0.707107i −0.866025 0.500000i 1.00000i 0.589284 0.157898i −0.965926 + 0.258819i −1.91156 1.82919i −0.707107 0.707107i 0.500000 + 0.866025i 0.305036 0.528338i
145.10 0.707107 0.707107i −0.866025 0.500000i 1.00000i 2.65393 0.711118i −0.965926 + 0.258819i 1.40673 2.24078i −0.707107 0.707107i 0.500000 + 0.866025i 1.37378 2.37945i
241.1 −0.707107 0.707107i −0.866025 + 0.500000i 1.00000i −4.23240 1.13407i 0.965926 + 0.258819i 1.34853 2.27629i 0.707107 0.707107i 0.500000 0.866025i 2.19085 + 3.79467i
241.2 −0.707107 0.707107i −0.866025 + 0.500000i 1.00000i −1.30432 0.349490i 0.965926 + 0.258819i −1.83449 + 1.90648i 0.707107 0.707107i 0.500000 0.866025i 0.675164 + 1.16942i
241.3 −0.707107 0.707107i −0.866025 + 0.500000i 1.00000i −0.981662 0.263036i 0.965926 + 0.258819i 2.09303 + 1.61840i 0.707107 0.707107i 0.500000 0.866025i 0.508146 + 0.880134i
241.4 −0.707107 0.707107i −0.866025 + 0.500000i 1.00000i 3.01304 + 0.807342i 0.965926 + 0.258819i 2.56888 0.633142i 0.707107 0.707107i 0.500000 0.866025i −1.55967 2.70142i
241.5 −0.707107 0.707107i −0.866025 + 0.500000i 1.00000i 3.50534 + 0.939253i 0.965926 + 0.258819i −2.61012 + 0.432735i 0.707107 0.707107i 0.500000 0.866025i −1.81450 3.14280i
241.6 0.707107 + 0.707107i −0.866025 + 0.500000i 1.00000i −2.31481 0.620250i −0.965926 0.258819i −2.57277 0.617120i −0.707107 + 0.707107i 0.500000 0.866025i −1.19823 2.07540i
241.7 0.707107 + 0.707107i −0.866025 + 0.500000i 1.00000i −1.42776 0.382567i −0.965926 0.258819i 0.349274 2.62260i −0.707107 + 0.707107i 0.500000 0.866025i −0.739062 1.28009i
241.8 0.707107 + 0.707107i −0.866025 + 0.500000i 1.00000i 0.499350 + 0.133800i −0.965926 0.258819i 0.430450 2.61050i −0.707107 + 0.707107i 0.500000 0.866025i 0.258483 + 0.447705i
241.9 0.707107 + 0.707107i −0.866025 + 0.500000i 1.00000i 0.589284 + 0.157898i −0.965926 0.258819i −1.91156 + 1.82919i −0.707107 + 0.707107i 0.500000 0.866025i 0.305036 + 0.528338i
241.10 0.707107 + 0.707107i −0.866025 + 0.500000i 1.00000i 2.65393 + 0.711118i −0.965926 0.258819i 1.40673 + 2.24078i −0.707107 + 0.707107i 0.500000 0.866025i 1.37378 + 2.37945i
See all 40 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 409.10 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.ba even 12 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 546.2.cg.b yes 40
7.d odd 6 1 546.2.by.b 40
13.f odd 12 1 546.2.by.b 40
91.ba even 12 1 inner 546.2.cg.b yes 40

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
546.2.by.b 40 7.d odd 6 1
546.2.by.b 40 13.f odd 12 1
546.2.cg.b yes 40 1.a even 1 1 trivial
546.2.cg.b yes 40 91.ba even 12 1 inner

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$20\!\cdots\!48$$$$T_{5}^{15} +$$$$42\!\cdots\!84$$$$T_{5}^{14} +$$$$56\!\cdots\!04$$$$T_{5}^{13} +$$$$52\!\cdots\!80$$$$T_{5}^{12} +$$$$30\!\cdots\!80$$$$T_{5}^{11} -$$$$11\!\cdots\!90$$$$T_{5}^{10} -$$$$39\!\cdots\!84$$$$T_{5}^{9} -$$$$33\!\cdots\!51$$$$T_{5}^{8} - 958938592008 T_{5}^{7} +$$$$26\!\cdots\!16$$$$T_{5}^{6} +$$$$21\!\cdots\!00$$$$T_{5}^{5} -$$$$11\!\cdots\!64$$$$T_{5}^{4} - 663279611904 T_{5}^{3} + 410533795584 T_{5}^{2} - 73784411136 T_{5} + 14841086976$$">$$T_{5}^{40} + \cdots$$ acting on $$S_{2}^{\mathrm{new}}(546, [\chi])$$.